I've been to both state and private. These are specifically my experiences, so may well be different for different people.
Firstly most private schools have no SEN (or very mild at best). So in my opinion they mostly have good exam results by fraud.
Private schools are very competitive (unlike state schools) - I always knew exactly where I was in the class and had a report with my place in each subject at the end of each term. If I was failing at something I would get extra encouragement. State schools only intervene if a pupil is failing badly because they just don't have the resources to waste on someone who's just coasting. That's where motivated parenting comes very handy.
All kids are encouraged to do as much as possible extracurricular stuff in private. For me this included things like qualifications in public speaking (in year 7 - 1st form), sports (eg horseriding) and music. We regularly got eg tickets to Wimbledon, and visits from sports stars and others at the top of their game who gave rousing speeches. That had to have had an impact.
State school is massively more inclusive, but can be very anti-competitive because they're trying to get everyone through as well as possible. There are a lot of extra-curricular activities in some state schools, but I got very little encouragement to do anything - so I largely didn't.
There are people on very little money in private schools, just as there are a lot of very wealthy parents whose children are in state schools. In both, wealth gives some advantage but doesn't account for everything.
People in private schools think kids in state schools are feral - I think a state school would have eaten my gentle DS alive. He might not have found an inspirational physics teacher. He wouldn't have found an environment where being very clever was kind of cool. They're not, they're just kids - some are gentle, some are academic. There are good teachers. I would say that the above was a ridiculous statement, but I understand that the fear is real, if unfounded.
It's amazingly common - the 7% who go to private schools think all the other 93% are somehow different from them. I heard that a lot at parties with my parents (luckily my parents didn't think that!).
Private school kids also learn to speak better. I learned to have a very clear, 'posh' speaking voice there which has helped me at times (especially when I'm phoning someone and want to sound like I have some authority.).